Poincaré duality and commutative differential graded algebras
[Dualité de Poincaré et algèbres différentielles graduées commutatives]
Lambrechts, Pascal ; Stanley, Don
Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008), p. 497-511 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous démontrons que toute algèbre différentielle graduée commutative (ADGC) dont la cohomologie est une algèbre simplement connexe à dualité de Poincaré est quasi-isomorphe à une ADGC dont l'algèbre sous-jacente est à dualité de Poincaré dans la même dimension. Ce résultat a des applications en théorie de l'homotopie rationnelle, permettant d'obtenir la dualité de Poincaré au niveau des cochaînes, entre autres dans l'étude des espaces de configurations et en topologie des cordes.

We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincaré duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincaré duality in the same dimension. This has applications in rational homotopy, giving Poincaré duality at the cochain level, which is of interest in particular in the study of configuration spaces and in string topology.

Publié le : 2008-01-01
DOI : https://doi.org/10.24033/asens.2074
Classification:  55P62,  55M05,  57P10
Mots clés: dualité de poincaré, algèbre différentielle commutative graduée 
@article{ASENS_2008_4_41_4_497_0,
     author = {Lambrechts, Pascal and Stanley, Don},
     title = {Poincar\'e duality and commutative differential graded algebras},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {41},
     year = {2008},
     pages = {497-511},
     doi = {10.24033/asens.2074},
     mrnumber = {2489632},
     zbl = {1172.13009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2008_4_41_4_497_0}
}

Lambrechts, Pascal; Stanley, Don. Poincaré duality and commutative differential graded algebras. Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008) pp. 497-511. doi : 10.24033/asens.2074. http://gdmltest.u-ga.fr/item/ASENS_2008_4_41_4_497_0/

[1] M. Aubry, J.-M. Lemaire & S. Halperin, Poincaré duality models, preprint.

[2] H. J. Baues, Algebraic homotopy, Cambridge Studies in Advanced Mathematics 15, Cambridge University Press, 1989. | MR 985099 | Zbl 0688.55001

[3] M. Chas & D. Sullivan, String topology, preprint arXiv:math/9911159, 1999.

[4] Y. Félix, S. Halperin & J.-C. Thomas, Rational homotopy theory, Graduate Texts in Math. 205, Springer, 2001. | Zbl 0961.55002

[5] Y. Félix & J.-C. Thomas, Rational BV-algebra in string topology, Bull. Soc. Math. France 136 (2008), 311-327. | Numdam | Zbl 1160.55006

[6] Y. Félix, J.-C. Thomas & M. Vigué-Poirrier, Rational string topology, J. Eur. Math. Soc. 9 (2007), 123-156. | Zbl 1200.55015

[7] W. Fulton & R. Macpherson, A compactification of configuration spaces, Ann. of Math. 139 (1994), 183-225. | Zbl 0820.14037

[8] I. Kříž, On the rational homotopy type of configuration spaces, Ann. of Math. 139 (1994), 227-237. | MR 1274092 | Zbl 0829.55008

[9] P. Lambrechts, Cochain model for thickenings and its application to rational LS-category, Manuscripta Math. 103 (2000), 143-160. | MR 1796311 | Zbl 0964.55013

[10] P. Lambrechts & D. Stanley, The rational homotopy type of configuration spaces of two points, Ann. Inst. Fourier (Grenoble) 54 (2004), 1029-1052. | Numdam | Zbl 1069.55006

[11] P. Lambrechts & D. Stanley, A remarkable DG-module model for configuration spaces, Algebraic & Geometric Topology 8 (2008), 1191-1222. | Zbl 1152.55004

[12] L. Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology, preprint arXiv:0711.1946, 2007. | Numdam | MR 2543477 | Zbl 1180.16007

[13] J. Milnor & D. Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer, 1973. | Zbl 0292.10016

[14] J. Neisendorfer & T. Miller, Formal and coformal spaces, Illinois J. Math. 22 (1978), 565-580. | Zbl 0396.55011

[15] J. Stasheff, Rational Poincaré duality spaces, Illinois J. Math. 27 (1983), 104-109. | MR 684544 | Zbl 0488.55010

[16] D. Sullivan, Infinitesimal computations in topology, Publ. Math. I.H.É.S. 47 (1977), 269-331. | Numdam | MR 646078 | Zbl 0374.57002

[17] T. Tradler, The BV algebra on Hochschild cohomology induced by infinity inner products, preprint arXiv:math.QA/0210150.

[18] T. Yang, A Batalin-Vilkovisky algebra structure on the Hochschild cohomology of truncated polynomials, Mémoire, University of Regina, 2007. | Zbl 1282.55013